Might be a silly question, or a complete lack of intuition from me, but I cannot figure out how to properly solve limits using some trivial variable change.
The example that triggered me is the limit you obtain while differentiating $e^x$:
$$\lim_{h \to 0} \frac{e^h - 1}{h} = 1$$
From here, one could introduce the variable change $u = e^h - 1 \implies h = \ln(u + 1)$ and then obtain $\lim_{u \to 0} \frac{u}{\ln(u + 1)} = 1$. Now, one could again multiply both the numerator and denominator by $\frac{1}{u}$:
$\lim_{u \to 0} \frac{u}{\ln(u + 1)} \cdot \frac{\frac{1}{u}}{\frac{1}{u}} = \lim_{u \to 0} \frac{1}{\ln(u + 1) \cdot \frac{1}{u}} = 1$
I am just trying to understand if there is an underlying formal framework behind those substitutions or it only involves intuitive pattern recognition.
It is mainly a matter of tricky pattern recognition.
Usually the foundamental result proved as first step is that as $x \to \infty \implies \left(1+\frac1x\right)^x\to e$ and from here we can prove that both
$$\lim_{h \to 0} \frac{e^h - 1}{h} = 1 \quad \lim_{h \to 0} \frac{\log (1+h)}{h} = 1$$
which are indeed equivalent.